The first is consider locally homogeneous Hermitian manifolds, namely, a compact Hermitian manifold \((M^n,g)\)  whose universal covering space is a homogeneous Hermitian manifold. A large and important subset of this is the so-called Lie-Hermitian manifolds, which means compact Hermitian manifold \((M^n,g)\)  whose universal covering space is a (connected, simply-connected, even-dimensional) Lie group \(G\)  equipped with a left invariant complex structure and a compatible left invariant metric. In a recent work [1]}, Y. Li and the second named author confirmed the conjecture for complex nilmanifolds, namely, Lie-Hermitian manifolds with \(G\)  nilpotent. For reasons explained in [1]}, we believe that one should explore other classes of Lie-Hermitian manifolds, or more generally locally homogeneous Hermitian manifolds, in hope of either proving the conjecture in the special case or producing counterexamples.
